Bear Put Spreads With Limited Risk

What a Bear Put Spread Does

A bear put spread is a two-option position designed to have a positive payoff when the underlying price falls, with that payoff capped once the price is sufficiently low. The spread buys one put with a higher strike and sells one put with a lower strike, using the same underlying and expiration. Its payoff depends on both options together: the short put reduces the payoff that the long put would have produced at low underlying prices.

This tutorial focuses on the options’ expiration payoff, before accounting for premiums paid or received when the position is opened. The distinction matters: a payoff describes what the options deliver at expiration; profit also depends on the initial cash flows and their timing.

Definition: Let $K_L$ be the lower strike and $K_H$ the higher strike, with $K_L<K_H$. A one-unit bear put spread consists of buying a put with strike $K_H$ and selling a put with strike $K_L$, both on the same underlying and with the same expiration. If the underlying price at expiration is $S_T$, its payoff per underlying unit is $\max(K_H-S_T,0)-\max(K_L-S_T,0)$.

Assume the options are European and are held to expiration. A put with strike $K$ pays $\max(K-S_T,0)$ per underlying unit. The long higher-strike put contributes a nonnegative payoff; the short lower-strike put contributes the negative of its own put payoff. No interest rate or probability model is needed to calculate these expiration payoffs. The options’ premiums, contract multiplier, and any transaction costs are not included unless the problem explicitly asks for those quantities.

Deriving the Payoff in Three Price Regions

The two strike prices divide possible expiration prices into three regions. In each region, evaluate the long put and short put separately, then add their signed payoffs. This approach prevents a common sign error: the payoff of a put that has been sold is subtracted, not added.

Below the lower strike.
If $S_T<K_L$, both puts have positive intrinsic value. The long put pays $K_H-S_T$; the short put contributes $-(K_L-S_T)$. Their sum is $K_H-K_L$.

Between the strikes.
If $K_L\leq S_T<K_H$, the higher-strike put has positive intrinsic value, while the lower-strike put has zero intrinsic value. The spread payoff is $K_H-S_T$.

At or above the higher strike.
If $S_T\geq K_H$, neither put has positive intrinsic value. The spread payoff is zero.

\begin{aligned} \text{Payoff} &=\max(K_H-S_T,0)-\max(K_L-S_T,0)\\ &= \begin{cases} K_H-K_L, & S_T<K_L,\\ K_H-S_T, & K_L\leq S_T<K_H,\\ 0, & S_T\geq K_H. \end{cases} \end{aligned}

The boundary cases fit the formula directly. At $S_T=K_L$, the lower-strike put is at the money and has zero intrinsic value, so the payoff is $K_H-K_L$. At $S_T=K_H$, the higher-strike put is at the money and has zero intrinsic value, so the spread payoff is zero. Writing the inequalities carefully avoids describing an at-the-money put as in the money or out of the money.

Derivation: For $S_T<K_L$, both positive-part expressions are active, so subtracting the short put’s payoff gives $(K_H-S_T)-(K_L-S_T)=K_H-K_L$. For $K_L\leq S_T<K_H$, the lower put payoff is zero and the higher put payoff is $K_H-S_T$. For $S_T\geq K_H$, both positive-part expressions are zero. These cases give the stated piecewise payoff, including the boundary values.

The formula shows the shape of the expiration payoff without needing a graph. It is constant below the lower strike, declines as $S_T$ rises through the interval between the strikes, and is zero at and above the higher strike. In particular, a further decline below $K_L$ does not increase the spread’s payoff: the long put’s additional payoff is offset by the additional loss on the short put.

Worked Examples

Worked Example: Evaluating Payoff at and Between the Strikes

A hypothetical investor buys a put with strike $K_H=\$60$ and sells a put with strike $K_L=\$50$. Both are European options on the same underlying, with the same expiration. Find the spread payoff per underlying unit if $S_T$ is $\$42$, $\$50$, $\$56$, $\$60$, or $\$67$.

Method. Apply the piecewise payoff formula to each price, keeping the short put’s payoff negative. The strike width is $K_H-K_L=\$60-\$50=\$10$.

At $S_T=\$42$, the price is below both strikes. The long put payoff is $\$60-\$42=\$18$, and the short put contribution is $-(\$50-\$42)=-\$8$. Thus the spread payoff is $\$18-\$8=\$10$.

At $S_T=\$50$, the lower-strike put is at the money and pays zero. The long put pays $\$60-\$50=\$10$, so the spread payoff is $\$10$.

At $S_T=\$56$, the price is between the strikes. The long put pays $\$60-\$56=\$4$, while the lower-strike put pays zero. The spread payoff is $\$4$.

At $S_T=\$60$, the higher-strike put is at the money and pays zero. The lower-strike put is also worth zero, so the spread payoff is $\$0$. At $S_T=\$67$, both puts have zero payoff, giving a spread payoff of $\$0$.

Check. The results are $\$10,\ \$10,\ \$4,\ \$0,$ and $\$0$, respectively. They agree with the three regions: the payoff is capped at the $\$10$ strike width below $K_L$, falls between the strikes, and is zero at and above $K_H$. The method avoids treating the lower put as an additional long position.

Worked Example: Scaling for Several Contracts

A hypothetical account buys three puts with strike $K_H=\$125$ and sells three puts with strike $K_L=\$110$. All options have the same expiration and underlying. Each contract has a multiplier of 100 underlying units. Calculate the total expiration payoff if the underlying finishes at $\$100$, $\$116$, $\$125$, or $\$140$.

Method. First calculate the payoff per underlying unit for one spread. Then multiply by three spreads and by 100 units per contract. The total scaling factor is $3\times100=300$.

At $S_T=\$100$, which is below $K_L$, one spread pays $K_H-K_L=\$125-\$110=\$15$ per unit. The account payoff is $\$15\times300=\$4{,}500$.

At $S_T=\$116$, which is between the strikes, one spread pays $K_H-S_T=\$125-\$116=\$9$ per unit. The account payoff is $\$9\times300=\$2{,}700$.

At $S_T=\$125$, the higher-strike put is at the money. Both put payoffs are zero, so the per-unit payoff and the account payoff are both $\$0$.

At $S_T=\$140$, both puts expire out of the money. The payoff is again $\$0$.

Check. The complete account payoffs are $\$4{,}500,\ \$2{,}700,\ \$0,$ and $\$0$. The first result also checks against the maximum possible per-unit payoff of the strike width, $\$15$, multiplied by 300 units. The multiplier scales the payoff; it does not change which price region applies.

Worked Example: Keeping Premium Separate From Payoff

A hypothetical trader buys a put with strike $K_H=\$80$ for a premium of $\$7.40$ per unit and sells a put with strike $K_L=\$70$ for a premium of $\$2.10$ per unit. The options share an expiration. Calculate the spread’s expiration payoff per unit at $S_T=\$68$, $\$75$, and $\$90$. Do not calculate profit.

Method. Use only the option payoffs at expiration. The premiums are initial cash flows, not terms in the payoff formula; they are supplied here to test that distinction.

At $S_T=\$68$, both puts have positive intrinsic value. The long put pays $\$80-\$68=\$12$. The short put contributes $-(\$70-\$68)=-\$2$. The spread payoff is $\$12-\$2=\$10$.

At $S_T=\$75$, only the higher-strike put has positive intrinsic value. Its payoff is $\$80-\$75=\$5$, and the lower-strike put payoff is zero. The spread payoff is $\$5$.

At $S_T=\$90$, both puts have zero intrinsic value. The spread payoff is $\$0$.

Check. The expiration payoffs are $\$10,\ \$5,$ and $\$0$. The premium difference is $\$7.40-\$2.10=\$5.30$ paid initially per unit, but subtracting $\$5.30$ from these amounts would change the question from payoff to a profit calculation. The distinction prevents an initial premium from being mistakenly inserted into the expiration payoff.

Position Checks and Common Traps

A reliable way to check a proposed bear put spread is to write down each leg with its sign before combining them:

\text{long put at }K_H:\ \max(K_H-S_T,0), \qquad \text{short put at }K_L:\ -\max(K_L-S_T,0).

The higher strike must belong to the purchased put and the lower strike to the sold put. Reversing either position changes the payoff shape. Also check that both options have the same underlying and expiration; otherwise the simple two-strike formula does not describe the combined payoff at one common expiration.

Common mistake: Suppose $K_H=\$45$, $K_L=\$35$, and $S_T=\$30$. An incorrect calculation might add both intrinsic values and report $(\$45-\$30)+(\$35-\$30)=\$20$. This treats both options as long puts. The lower-strike put was sold, so its payoff must be subtracted: $(\$45-\$30)-(\$35-\$30)=\$15-\$5=\$10$. That equals the strike width, $\$45-\$35=\$10$, as it should below the lower strike.

Another error is to use weak inequalities carelessly in verbal descriptions. If $S_T=K_L$, the lower-strike put is at the money, not in the money; its payoff is zero. If $S_T=K_H$, the higher-strike put is at the money, not out of the money; its payoff is also zero. The formula remains valid at both boundaries, but describing those options accurately helps prevent mistakes when moving from a verbal condition to a calculation.

Key takeaway: A bear put spread buys the higher-strike put and sells the lower-strike put. Its payoff per unit is the difference of the long put payoff and the lower put payoff. Below the lower strike it equals the strike width; between the strikes it equals $K_H-S_T$; at or above the higher strike it is zero. Premiums and other initial cash flows are not part of this expiration-payoff calculation.

Check Your Understanding

For each problem, identify the two legs and apply the expiration-payoff formula. Answers are per underlying unit unless a contract multiplier is specified.

A spread buys a put at $K_H=\$95$ and sells a put at $K_L=\$80$. What is its payoff at $S_T=\$72$, and which strike-region rule applies? (Answer: $\$15$; below the lower strike.)
A spread buys the $K_H=\$140$ put and sells the $K_L=\$125$ put. Find the payoff at $S_T=\$132$, and state the payoff of each leg. (Answer: Long put $\$8$, short put $\$0$; spread payoff $\$8$.)
A spread buys the $K_H=\$62$ put and sells the $K_L=\$47$ put. Evaluate the spread exactly at $S_T=\$47$, identifying the status and payoff of the lower-strike put. (Answer: The lower-strike put is at the money and pays $\$0$; spread payoff $\$15$.)
An account buys two $K_H=\$210$ puts and sells two $K_L=\$190$ puts. Each option contract represents 50 units. Find the account payoff at $S_T=\$198$. (Answer: Per spread, $\$210-\$198=\$12$; total payoff $2\times50\times\$12=\$1{,}200$.)
A trader buys a $K_H=\$105$ put for $\$9$ and sells a $K_L=\$90$ put for $\$3$, with the same expiration. Find the expiration payoff at $S_T=\$84$, and explain whether the $\$6$ premium difference belongs in that payoff. (Answer: $(\$105-\$84)-(\$90-\$84)=\$21-\$6=\$15$; the premium difference is excluded from expiration payoff.)

What You'll Learn